In a railway system, and especially in a high-density commuter or subway systems, trains run along a route according to a schedule that can have different travel times that arise from an overall schedule, which is usually managed by a control center. Speeds are constrained by planned and unplanned events, such as maintenance accidents, passenger density, weather, and the like. Thus, it is necessary to determine an optimal run-curve for the train according to dynamic travel time requirement and speed limit constraints. The run-curve profile can be optimized such that energy consumption is minimized, while simultaneously satisfying all constraints of Motion, such as speed limits, safety zones, and etc. Efficient run-curves for vehicles can reduce energy consumption.
In the railway system, the trains can be equipped with regenerative brakes, batteries, and other traction and energy transformation devices. A topology of the system is fixed. The topology reflects the lengths, run-curves, and slope of the various routes. A resistance from air and tracks can also be a function of the speed and location of the train along the route. At a large scale, the mass of the train is relatively constant.
The travel times, subject to preplanned schedules and dynamic events, often are unknown until just before departure, and in some cases, along the route. Thus, it is required to optimize the run-curves in real-time.
Dynamics of the vehicle can be described by
                                                        ⅆ              v                                      ⅆ              t                                =                      a            ⁡                          [                                                z                  ⁡                                      (                    t                    )                                                  ,                                  v                  ⁡                                      (                    t                    )                                                  ,                                  u                  ⁡                                      (                    t                    )                                                              ]                                      ,                            (        1        )                                                                    ⅆ              z                                      ⅆ              t                                =                      v            ⁡                          (              t              )                                      ,                            (        2        )            where t, z, v, and u respectively represent time, location, velocity and action. Actions can include acceleration, deceleration, braking, and coasting. Other factors that can be considered can include air resistance, track resistance, track slope, motor efficiency, brake efficiency, and the like.
A vehicle rate of energy consumption E is
                              E          =                                    ∫              0              T                        ⁢                                          p                ⁡                                  [                                                            z                      ⁡                                              (                        t                        )                                                              ,                                          v                      ⁡                                              (                        t                        )                                                              ,                                          u                      ⁡                                              (                        t                        )                                                                              ]                                            ⁢                                                          ⁢                              ⅆ                t                                                    ,                            (        3        )            where T is the travel time. The power consumption rate p at time t depends on corresponding vehicle location, speed, and action. The function p returns the rate of energy consumption integrated over the travel time, which is related to state of the vehicle, and action. For a complicate statement of the energy consumption rate, it is better to express the rate as a function of p with all factors as inputs. Under other assumption, p can have different forms, but the function form is more general.
Run-curve optimization is a minimization problem that uses an objective functionJ=μE+(1−μ)T  (4)subject to the constraints in equations (1-3), where a weight μ describes a relative importance of minimizing the travel time with respect the rate of energy consumption.
A number of prior art methods for solving the minimization optimization problem are known, such as dynamic programming, heuristic optimization, genetic algorithms, and nonlinear optimization. However, those methods can be suboptimal or computationally complex precluding real-time solutions.